The Number Of Effective Modes Of The Two-Dimensional And 3-Dimensional Nonlinear Schrodinger-Equation
Abbreviated Journal Title
Phys. Rev. A
In this paper we give a discussion of the number of “effective modes” of the two- and three-dimensional nonlinear Schrödinger equation. First, we give a simple analytic explanation of the numerical results of Martin and Yuen on the long-time evolution of spatially periodic solutions of the two-dimensional nonlinear Schrödinger equation for a weakly nonlinear, deep-water, modulated gravity wavetrain. We show that this nonlinear system can be described as effectively possessing an arbitrarily high number of degrees of freedom so that the overall motion can at best be only quasi-recurrent. Then, we explore the connection between the number of “effective modes” and the collapsing tendency of solutions of the spherically-symmetric nonlinear Schrödinger equation. In the three-dimensional cubic nonlinear case where the solutions are known to exhibit collapsing tendency, we find that the number of “effective modes” increases with the increase in the wave-amplitude (this is in contrast to the one-dimensional case). We give quantitative estimates of the limitations of the cubic nonlinear model, nonetheless, in describing the Langmuir collapse process. In the three-dimensional full exponential nonlinear case where the solutions are known to exhibit no collapsing tendency and evolve into localized stationary structures, we find a clear reduction in the number of “effective modes”. Indeed, the number is independent of the magnitude of the wave amplitude! These results appear to indicate that the number of “effective modes” can also serve as a measure of the collapsing tendency of the solutions of the spherically-symmetric nonlinear Schrödinger equation.
Physical Review A
Shivamoggi, Bhimsen K. and Mohapatra, Ram N., "The Number Of Effective Modes Of The Two-Dimensional And 3-Dimensional Nonlinear Schrodinger-Equation" (1988). Faculty Bibliography 1980s. 702.